For a continuous function $f=\left(f_{1}, f_{2}, \ldots, f_{m}\right):[0,1] \rightarrow \mathbb{R}^{m}$, define

$$\int_{0}^{1} f(t) d t=\left(\int_{0}^{1} f_{1}(t) d t, \int_{0}^{1} f_{2}(t) d t, \ldots, \int_{0}^{1} f_{m}(t) d t\right)$$

Show that

$$\left\|\int_{0}^{1} f(t) d t\right\|_{2} \leqslant \int_{0}^{1}\|f(t)\|_{2} d t$$

for every continuous function $f:[0,1] \rightarrow \mathbb{R}^{m}$, where $\|\cdot\|_{2}$ denotes the Euclidean norm on $\mathbb{R}^{m}$.

Find all continuous functions $f:[0,1] \rightarrow \mathbb{R}^{m}$ with the property that

$$\left\|\int_{0}^{1} f(t) d t\right\|=\int_{0}^{1}\|f(t)\| d t$$

regardless of the norm $\|\cdot\|$ on $\mathbb{R}^{m}$.

[Hint: start by analysing the case when $\|\cdot\|$ is the Euclidean norm $\|\cdot\|_{2}$.]